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Theorem maduval 18577
Description: Second substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
Hypotheses
Ref Expression
madufval.a  |-  A  =  ( N Mat  R )
madufval.d  |-  D  =  ( N maDet  R )
madufval.j  |-  J  =  ( N maAdju  R )
madufval.b  |-  B  =  ( Base `  A
)
madufval.o  |-  .1.  =  ( 1r `  R )
madufval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
maduval  |-  ( M  e.  B  ->  ( J `  M )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
Distinct variable groups:    i, N, j, k, l    R, i, j, k, l    i, M, j, k, l
Allowed substitution hints:    A( i, j, k, l)    B( i, j, k, l)    D( i, j, k, l)    .1. ( i, j, k, l)    J( i, j, k, l)    .0. ( i, j, k, l)

Proof of Theorem maduval
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 madufval.a . . . . 5  |-  A  =  ( N Mat  R )
2 madufval.b . . . . 5  |-  B  =  ( Base `  A
)
31, 2matrcl 18438 . . . 4  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 459 . . 3  |-  ( M  e.  B  ->  N  e.  Fin )
5 eqid 2454 . . . 4  |-  ( i  e.  N ,  j  e.  N  |->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
65mpt2exg 6759 . . 3  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e.  _V )
74, 4, 6syl2anc 661 . 2  |-  ( M  e.  B  ->  (
i  e.  N , 
j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e. 
_V )
8 oveq 6207 . . . . . . . . 9  |-  ( m  =  M  ->  (
k m l )  =  ( k M l ) )
98ifeq2d 3917 . . . . . . . 8  |-  ( m  =  M  ->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) )  =  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
1093ad2ant1 1009 . . . . . . 7  |-  ( ( m  =  M  /\  k  e.  N  /\  l  e.  N )  ->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) )  =  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
1110mpt2eq3dva 6260 . . . . . 6  |-  ( m  =  M  ->  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) )  =  ( k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
12113ad2ant1 1009 . . . . 5  |-  ( ( m  =  M  /\  i  e.  N  /\  j  e.  N )  ->  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
1312fveq2d 5804 . . . 4  |-  ( ( m  =  M  /\  i  e.  N  /\  j  e.  N )  ->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
1413mpt2eq3dva 6260 . . 3  |-  ( m  =  M  ->  (
i  e.  N , 
j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
15 madufval.d . . . 4  |-  D  =  ( N maDet  R )
16 madufval.j . . . 4  |-  J  =  ( N maAdju  R )
17 madufval.o . . . 4  |-  .1.  =  ( 1r `  R )
18 madufval.z . . . 4  |-  .0.  =  ( 0g `  R )
191, 15, 16, 2, 17, 18madufval 18576 . . 3  |-  J  =  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )
2014, 19fvmptg 5882 . 2  |-  ( ( M  e.  B  /\  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e.  _V )  -> 
( J `  M
)  =  ( i  e.  N ,  j  e.  N  |->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
217, 20mpdan 668 1  |-  ( M  e.  B  ->  ( J `  M )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   ifcif 3900   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203   Fincfn 7421   Basecbs 14293   0gc0g 14498   1rcur 16726   Mat cmat 18406   maDet cmdat 18523   maAdju cmadu 18571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-slot 14297  df-base 14298  df-mat 18408  df-madu 18573
This theorem is referenced by:  maducoeval  18578
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